Thursday, May 19, 2016

Physics and Art: Pollock

This winter, one of my students came to me with an idea to create an 'art robot' - a robot that could create a piece of art. After some research, she found a Discover article summarizing Richard Taylor's work on fractal analysis of Jackson Pollock's drip paintings. Note that there is considerable disagreement over whether this is valid, as the range of available scales (only about three orders of magnitude) is generally considered too small to adequately authenticate fractal structure.

This hiccough notwithstanding, we set about to create a chaotic pendulum that could create paintings with similar fractal dimensions to Pollock's. The general idea of structural complexity is shown in a graphic from the first reference:

The fractal dimension measured by Taylor of Pollock's works is in the middle range: 1.5-1.6. A non-chaotic conical pendulum's traces are not complex enough to reproduce this (left image below), but a chaotic pendulum can (middle and right below, from second reference above):
In order to make the traces chaotic, we attached a pulley controlled by a stepper motor to the pivot of the pendulum and had an Arduino quickly oscillate the pivot infrequently (slightly lower period than the pendulum's natural period). A Glowscript simulation of two such pendula starting with nearly identical initial conditions show quick divergence, the result of the nonlinear chaotic system.

In practice, we gave the system some random horizontal driving as well, as it lost amplitude. The result is a set of five attractive canvases. The set will adorn the hall near my room, along with a poster explaining the process, showing the derivation, and discussing the results (I co-opted some MATLAB code to determine the fractal dimension of each foreground color-layer of each painting and each painting's average fractal dimension).

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